Dynamic Causal Modelling (DCM) for fMRI Klaas Enno Stephan Laboratory for Social & Neural Systems Research (SNS) University of Zurich Wellcome Trust Centre for Neuroimaging University College London SPM Course, FIL 13 May 2011

Structural, functional & effective connectivity anatomical/structural connectivity = presence of axonal connections functional connectivity = statistical dependencies between regional time series effective connectivity = directed influences between neurons or neuronal populations Sporns 2007, Scholarpedia

Some models of effective connectivity for fMRI data Structural Equation Modelling (SEM) McIntosh et al. 1991, 1994; Büchel & Friston 1997; Bullmore et al. 2000 regression models (e.g. psycho-physiological interactions, PPIs) Friston et al. 1997 Volterra kernels Friston & Büchel 2000 Time series models (e.g. MAR/VAR, Granger causality) Harrison et al. 2003, Goebel et al. 2003 Dynamic Causal Modelling (DCM) bilinear: Friston et al. 2003; nonlinear: Stephan et al. 2008

Dynamic causal modelling (DCM) DCM framework was introduced in 2003 for fMRI by Karl Friston, Lee Harrison and Will Penny (NeuroImage 19:1273-1302) part of the SPM software package currently more than 160 published papers on DCM

Dynamic Causal Modeling (DCM) Hemodynamic forward model: neural activityBOLD Electromagnetic forward model: neural activityEEG MEG LFP Neural state equation: fMRI EEG/MEG simple neuronal model complicated forward model complicated neuronal model simple forward model inputs

Example: a linear model of interacting visual regions FG left FG right LG = lingual gyrus FG = fusiform gyrus Visual input in the - left (LVF) - right (RVF) visual field. x3 x4 LG left LG right x1 x2 RVF LVF u2 u1

Example: a linear model of interacting visual regions FG left FG right LG = lingual gyrus FG = fusiform gyrus Visual input in the - left (LVF) - right (RVF) visual field. x3 x4 LG left LG right x1 x2 RVF LVF u2 u1 state changes effective connectivity system state input parameters external inputs

Extension: bilinear model FG left FG right x3 x4 LG left LG right x1 x2 RVF CONTEXT LVF u2 u3 u1

   BOLD y y y y λ x neuronal states hemodynamic model activity x2(t) activity x3(t) activity x1(t) x neuronal states modulatory input u2(t) t integration endogenous connectivity direct inputs modulation of connectivity Neural state equation t driving input u1(t)

Bilinear DCM Two-dimensional Taylor series (around x0=0, u0=0): driving input modulation Two-dimensional Taylor series (around x0=0, u0=0): Bilinear state equation:

DCM parameters = rate constants Integration of a first-order linear differential equation gives an exponential function: Coupling parameter a is inversely proportional to the half life  of z(t): The coupling parameter a thus describes the speed of the exponential change in x(t)

Example: context-dependent decay u 1 Z 2 u1 stimuli u1 context u2 u2 - + - x1 x1 + x2 + x2 - - Penny et al. 2004, NeuroImage

The problem of hemodynamic convolution Goebel et al. 2003, Magn. Res. Med.

Hemodynamic forward models are important for connectivity analyses of fMRI data Granger causality DCM David et al. 2008, PLoS Biol.

The hemodynamic model in DCM stimulus functions u t neural state equation hemodynamic state equations Balloon model BOLD signal change equation The hemodynamic model in DCM Stephan et al. 2007, NeuroImage

How interdependent are neural and hemodynamic parameter estimates? B C h ε Stephan et al. 2007, NeuroImage

DCM is a Bayesian approach new data prior knowledge posterior  likelihood ∙ prior Bayes theorem allows one to formally incorporate prior knowledge into computing statistical probabilities. In DCM: empirical, principled & shrinkage priors. The “posterior” probability of the parameters given the data is an optimal combination of prior knowledge and new data, weighted by their relative precision.

modelled BOLD response stimulus function u Overview: parameter estimation neural state equation Combining the neural and hemodynamic states gives the complete forward model. An observation model includes measurement error e and confounds X (e.g. drift). Bayesian inversion: parameter estimation by means of variational EM under Laplace approximation Result: Gaussian a posteriori parameter distributions, characterised by mean ηθ|y and covariance Cθ|y. parameters hidden states state equation ηθ|y observation model modelled BOLD response

VB in a nutshell (mean-field approximation)  Neg. free-energy approx. to model evidence.  Mean field approx.  Maximise neg. free energy wrt. q = minimise divergence, by maximising variational energies  Iterative updating of sufficient statistics of approx. posteriors by gradient ascent.

Inference about DCM parameters: Bayesian single-subject analysis Gaussian assumptions about the posterior distributions of the parameters posterior probability that a certain parameter (or contrast of parameters cT ηθ|y) is above a chosen threshold γ: By default, γ is chosen as zero ("does the effect exist?").

Bayesian single subject inference LD|LVF p(cT>0|y) = 98.7% 0.13  0.19 0.34  0.14 FG left FG right LD LD 0.44  0.14 0.29  0.14 LG left LG right 0.01  0.17 -0.08  0.16 RVF stim. LD|RVF LVF stim. Contrast: Modulation LG right  LG links by LD|LVF vs. modulation LG left  LG right by LD|RVF Stephan et al. 2005, Ann. N.Y. Acad. Sci.

Inference about DCM parameters: Bayesian parameter averaging (FFX group analysis) Likelihood distributions from different subjects are independent  one can use the posterior from one subject as the prior for the next Under Gaussian assumptions this is easy to compute: group posterior covariance individual posterior covariances group posterior mean individual posterior covariances and means “Today’s posterior is tomorrow’s prior”

Inference about DCM parameters: RFX group analysis (frequentist) In analogy to “random effects” analyses in SPM, 2nd level analyses can be applied to DCM parameters: Separate fitting of identical models for each subject Selection of (bilinear) parameters of interest one-sample t-test: parameter > 0 ? paired t-test: parameter 1 > parameter 2 ? rmANOVA: e.g. in case of multiple sessions per subject

definition of model space inference on model structure or inference on model parameters? inference on individual models or model space partition? inference on parameters of an optimal model or parameters of all models? optimal model structure assumed to be identical across subjects? comparison of model families using FFX or RFX BMS optimal model structure assumed to be identical across subjects? BMA yes no yes no FFX BMS RFX BMS FFX BMS RFX BMS FFX analysis of parameter estimates (e.g. BPA) RFX analysis of parameter estimates (e.g. t-test, ANOVA) Stephan et al. 2010, NeuroImage

Any design that is good for a GLM of fMRI data. What type of design is good for DCM? Any design that is good for a GLM of fMRI data.

GLM vs. DCM DCM tries to model the same phenomena (i.e. local BOLD responses) as a GLM, just in a different way (via connectivity and its modulation). No activation detected by a GLM → no motivation to include this region in a deterministic DCM. However, a stochastic DCM could be applied despite the absence of a local activation. Stephan 2004, J. Anat.

Multifactorial design: explaining interactions with DCM Task factor Task A Task B Stim 1 Stim 2 Stimulus factor TA/S1 TB/S1 TA/S2 TB/S2 X1 X2 Stim2/ Task A Stim1/ Task A Stim 1/ Task B Stim 2/ GLM Let’s assume that an SPM analysis shows a main effect of stimulus in X1 and a stimulus  task interaction in X2. How do we model this using DCM? X1 X2 Stim2 Stim1 Task A Task B DCM

Simulated data X1 Stimulus 1 X1 X2 Stimulus 2 Task A Task B X2 – – +++ Stim 1 Task A Stim 2 Task A Stim 1 Task B Stim 2 Task B +++ Stimulus 2 + +++ + Task A Task B X2 Stephan et al. 2007, J. Biosci.

plus added noise (SNR=1) X1 Stim 1 Task A Stim 2 Task A Stim 1 Task B Stim 2 Task B X2 plus added noise (SNR=1)

DCM10 in SPM8 DCM10 was released as part of SPM8 in July 2010 (version 4010). Introduced many new features, incl. two-state DCMs and stochastic DCMs This led to various changes in model defaults, e.g. inputs mean-centred changes in coupling priors self-connections: separately estimated for each area For details, see: www.fil.ion.ucl.ac.uk/spm/software/spm8/SPM8_Release_Notes_r4010.pdf Further changes in version 4290 (released April 2011) to accommodate new developments and give users more choice (e.g. whether or not to mean- centre inputs).

The evolution of DCM in SPM DCM is not one specific model, but a framework for Bayesian inversion of dynamic system models The default implementation in SPM is evolving over time better numerical routines for inversion change in priors to cover new variants (e.g., stochastic DCMs, endogenous DCMs etc.) To enable replication of your results, you should ideally state which SPM version you are using when publishing papers.

Factorial structure of model specification in DCM10 Three dimensions of model specification: bilinear vs. nonlinear single-state vs. two-state (per region) deterministic vs. stochastic Specification via GUI.

bilinear DCM non-linear DCM driving input modulation non-linear DCM driving input modulation Two-dimensional Taylor series (around x0=0, u0=0): Bilinear state equation: Nonlinear state equation:

Nonlinear dynamic causal model (DCM) Neural population activity fMRI signal change (%) u2 x1 x2 x3 u1 Nonlinear dynamic causal model (DCM) Stephan et al. 2008, NeuroImage

attention PPC stim V1 V5 motion MAP = 1.25 0.10 PPC 0.26 0.39 1.25 0.26 stim V1 0.13 V5 0.46 0.50 motion Stephan et al. 2008, NeuroImage

motion & attention motion & no attention static dots V1 V5 PPC observed fitted

Extrinsic (between-region) coupling Intrinsic (within-region) coupling Two-state DCM Single-state DCM Two-state DCM input Extrinsic (between-region) coupling Intrinsic (within-region) coupling Marreiros et al. 2008, NeuroImage

Stochastic DCM accounts for stochastic neural fluctuations can be fitted to resting state data  has unknown precision and smoothness  additional hyperparameters Friston et al. (2008, 2011) NeuroImage Daunizeau et al. (2009) Physica D Li et al. (2011) NeuroImage

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